Discrete-time Nonlinear Stochastic Systems Research Articles

Passive estimated state feedback fuzzy controller design for discrete perturbed fuzzy systems with multiplicative noises

This article presents an estimated state feedback fuzzy controller design method for uncertain passive discrete-time nonlinear stochastic systems with multiplicative noises. The nonlinear stochastic systems considered in this article are represented by Takagi–Sugeno (T–S) fuzzy models. For describing stochastic behaviors, stochastic differential equations are used to structure the stochastic T–S fuzzy model for representing nonlinear stochastic systems. Besides, the uncertainties of the controlled system are considered for dealing with molding errors and varying parameters. The concept of parallel distributed compensation is employed in this article to construct the estimated state feedback fuzzy controllers. Applying the Lyapunov and passivity theories, the sufficient stability conditions are derived in terms of linear matrix inequality. Finally, a numerical example is provided to show the effectiveness and applicability of the proposed fuzzy controller design approach.

Imperfect premise matching based fuzzy control with passive constraints for discrete time-delay multiplicative noised stochastic nonlinear systems

This paper investigates a fuzzy controller design method for discrete-time nonlinear stochastic time-delay systems which are presented by the Takagi-Sugeno (T-S) fuzzy model with multiplicative noises. Utilizing the proposed design method, the fuzzy controller can be carried out via not only state feedback scheme but also output feedback scheme. Both of them are accomplished by the concept of imperfect premise matching (IPM). For discussing the stabilization problem, the Lyapunov-Krasovskii function and passivity theory are applied to derive the sufficient conditions. Moreover, the discrete Jensen inequality is employed to decrease the conservatism of the proposed method. Finally, a numerical example for the control of a nonlinear time-delay pendulum system is provided to show the effectiveness and usefulness of the proposed design method.

A class of robust bounded controllers tracking a nonlinear discrete-time stochastic system: Attractive ellipsoid technique application

This paper deals with designing of robust tracking controllers for a class of nonlinear uncertain discrete-time stochastic systems which are affine with respect to a control action on a bounded energy level. The considered nonlinear dynamics is admitted to be a priori unknown but belonging to the class of the so-called quasi-Lipschitz vector-field. The current states may be unavailable on-line, but the corresponding output vector is assumed to be measured during the process. Both the state dynamics and the output measurements are disturbed by external additive noises which are also supposed to be immeasurable. In this situation any suitable controller can only provide the boundedness of the tracking-error trajectories within a bounded zone with probability one. In this paper we suggest designing of both the control and observer “optimal” gain-matrices minimizing the “size” of the attractive ellipsoid containing all tracking-error trajectories in the vicinity of the origin. It is shown that this design problem of an output bounded control may be converted into the corresponding attractive ellipsoid “minimization” under some constraints of BMI's (bilinear matrix inequalities) type. The application of an adequate coordinate changing transforms these BMI's into a set of LMI's (linear matrix inequalities) that permits to use directly the standard MATLAB-toolbox. Two illustrative examples are considered: the first one concerns a two state—single output stochastic model, and the second one deals with a discrete-time model of electric-magnetic-tape-drive containing the four states-positions and two measured outputs.

Probabilistic DHP adaptive critic for nonlinear stochastic control systems

Following the recently developed algorithms for fully probabilistic control design for general dynamic stochastic systems (Herzallah & Káarnáy, 2011; Kárný, 1996), this paper presents the solution to the probabilistic dual heuristic programming (DHP) adaptive critic method (Herzallah & Káarnáy, 2011) and randomized control algorithm for stochastic nonlinear dynamical systems. The purpose of the randomized control input design is to make the joint probability density function of the closed loop system as close as possible to a predetermined ideal joint probability density function. This paper completes the previous work (Herzallah & Káarnáy, 2011; Kárný, 1996) by formulating and solving the fully probabilistic control design problem on the more general case of nonlinear stochastic discrete time systems. A simulated example is used to demonstrate the use of the algorithm and encouraging results have been obtained.

Modified state prediction algorithm based on UKF

The state prediction based on the unscented Kalman filter (UKF) for nonlinear stochastic discrete-time systems with linear measurement equation is investigated. Predicting future states by using the information of available measurements is an effective method to solve time delay problems. It not only helps the system operator to perform security analysis, but also allows more time for operator to take better decision in case of emergency. In addition, predictive state can make the system implement real-time monitoring and achieve good robustness. UKF has been popular in state prediction because of its advantages in handling nonlinear systems. However, the accuracy of prediction degrades notably once a filter uses a much longer future prediction. A confidence interval (CI) is proposed to overcome the problem. The advantages of CI are that it provides the information about states coverage, which is useful for treatment-plan evaluation, and it can be directly used to specify the margin to accommodate prediction errors. Meanwhile, the CI of prediction errors can be used to correct the predictive state, and thereby it improves the prediction accuracy. Simulations are provided to demonstrate the effectiveness of the theoretical results.

Deconvolution Filtering for Nonlinear Stochastic Systems with Randomly Occurring Sensor Delays via Probability-Dependent Method

This paper deals with a robustH∞deconvolution filtering problem for discrete-time nonlinear stochastic systems with randomly occurring sensor delays. The delayed measurements are assumed to occur in a random way characterized by a random variable sequence following the Bernoulli distribution with time-varying probability. The purpose is to design anH∞deconvolution filter such that, for all the admissible randomly occurring sensor delays, nonlinear disturbances, and external noises, the input signal distorted by the transmission channel could be recovered to a specified extent. By utilizing the constructed Lyapunov functional relying on the time-varying probability parameters, the desired sufficient criteria are derived. The proposedH∞deconvolution filter parameters include not only the fixed gains obtained by solving a convex optimization problem but also the online measurable time-varying probability. When the time-varying sensor delays occur randomly with a time-varying probability sequence, the proposed gain-scheduled filtering algorithm is very effective. The obtained design algorithm is finally verified in the light of simulation examples.

Stability of Nonlinear Stochastic Discrete-Time Systems

This paper studies the stability for nonlinear stochastic discrete-time systems. First of all, several definitions on stability are introduced, such as stability, asymptotical stability, andpth moment exponential stability. Moreover, using the method of the Lyapunov functionals, some efficient criteria for stochastic stability are obtained. Some examples are presented to illustrate the effectiveness of the proposed theoretical results.

Robust stability and stabilization of a class of nonlinear discrete time stochastic systems: An LMI approach

A problem of robust state feedback stability and stabilization of nonlinear discrete-time stochastic processes is considered. The linear rate vector of a discrete-time system is perturbed by a nonlinear function that satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of nonlinear discrete-time systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The state dependent diffusion is modeled by a normal sequence of identically independently distributed random variables. The new formulation provides a suitable setting for robust stabilization of nonlinear discrete-time systems where the underlying deterministic systems satisfy the generalized matching conditions. Our method which is based on linear matrix inequalities (LMIs) is distinctive from the existing robust control and absolute stability techniques. Examples are given to demonstrate the obtained results.

Lyapunov-based adaptive state estimation for a class of nonlinear stochastic systems

This paper is concerned with an adaptive state estimation problem for a class of nonlinear stochastic systems with unknown constant parameters. These nonlinear systems have a linear-in-parameter structure, and the nonlinearity is assumed to be bounded in a Lipschitz-like manner. Using stochastic counterparts of Lyapunov stability theory, we present adaptive state and parameter estimators with ultimately exponentially bounded estimator errors in the sense of mean square for both continuous-time and discrete-time nonlinear stochastic systems. Sufficient conditions are given in terms of the solvability of LMIs. Moreover, we also introduce a suboptimal design approach to optimizing the upper bound of the mean-square error of parameter estimation. This suboptimal design procedure is also realized by LMI computations. By a martingale method, we also show that the related Lyapunov function has a non-negative Lyapunov exponent.

First Passage Problems for Nonstationary Discrete-Time Stochastic Control Systems

This paper concerns first passage problems for nonstationary nonlinear discrete-time stochastic control systems. The state and control spaces are general Borel spaces, the transition probabilities are nonstationary, the costs/rewards are time-varying and may be unbounded. The optimal control problem is to minimize the expected discounted costs incurred until the first passage time to some target set, in which the discount factors are allowed to be both time- and state-dependent. Under reasonably mild conditions we establish the so-called first passage optimality equations, and prove that the optimal cost/reward functions satisfy the optimality equations. Furthermore, from the optimality equations we show the existence of optimal Markov policies.

A Fuzzy Approach to Robust Control of Stochastic Nonaffine Nonlinear Systems

This paper investigates the stabilization problem for a class of discrete‐time stochastic non‐affine nonlinear systems based on T‐S fuzzy models. Based on the function approximation capability of a class of stochastic T‐S fuzzy models, it is shown that the stabilization problem of a stochastic non‐affine nonlinear system can be solved as a robust stabilization problem of the stochastic T‐S fuzzy system with the approximation errors as the uncertainty term. By using a class of piecewise dynamic feedback fuzzy controllers and piecewise quadratic Lyapunov functions, robust semiglobal stabilization condition of the stochastic non‐affine nonlinear systems is formulated in terms of linear matrix inequalities. A simulation example illustrating the effectiveness of the proposed approach is provided in the end.

Towards unknown input filters for non-linear stochastic systems without a fault decoupling effect

The paper deals with the problem of designing filters for non-linear discrete-time stochastic systems. In particular, it is shown how to design an unknown input filter for a single (constant) unknown input distribution matrix, which guarntees that the effect of a fault will not be decoupled from the residual. Subsequently, the problem of using one, fixed disturbance distribution matrix is eliminated by using the interacting multiple models algorithm to select an appropriate unknown input distribution matrix from a predefined set of matrices. The final part of the paper shows an illustrative example, which confirms the effectiveness of the proposed approach. Copyright©2012 IFAC

Fuzzy control with robust and passive properties for discrete-time Takagi–Sugeno fuzzy systems with multiplicative noises

This paper deals with the problem of robust fuzzy control for uncertain passive discrete-time non-linear stochastic systems with multiplicative noise. In this paper the non-linear stochastic system is represented by a Takagi–Sugeno fuzzy model. Using that approach the concept of parallel distributed compensation is applied to construct the fuzzy controller. Sufficient stability conditions are derived in terms of a linear matrix inequality by applying the Lyapunov theory. In addition, the passivity theory and robust technique are used to guarantee strict input passivity and robust asymptotical stability of the closed-loop system with admissible parameter uncertainties. Finally, a numerical example concerning the control of a truck–trailer system is provided to demonstrate the validity of the proposed fuzzy controller design method.

An integrated optimal control algorithm for discrete-time nonlinear stochastic system

Consider a discrete-time nonlinear system with random disturbances appearing in the real plant and the output channel where the randomly perturbed output is measurable. An iterative procedure based on the linear quadratic Gaussian optimal control model is developed for solving the optimal control of this stochastic system. The optimal state estimate provided by Kalman filtering theory and the optimal control law obtained from the linear quadratic regulator problem are then integrated into the dynamic integrated system optimisation and parameter estimation algorithm. The iterative solutions of the optimal control problem for the model obtained converge to the solution of the original optimal control problem of the discrete-time nonlinear system, despite model-reality differences, when the convergence is achieved. An illustrative example is solved using the method proposed. The results obtained show the effectiveness of the algorithm proposed.

On parameter estimation of partly observed bilinear discrete-time stochastic systems

The parameter estimation problem of a partly observed nonlinear discrete-time stochastic system is considered. The unobserved component of the system is a q-dimensional stable autoregressive process of the pth order with random parameters, observed in the presence of multiplicative and additive noises. The distributions of all the noises of the system are supposed to be unknown. The problem is to estimate the mean of the drifting parameters of the object and variances of the additive noises of the system. Asymptotic correlation estimators of all these parameters are investigated and sequential estimators with given mean square accuracy of the mean of the drifting autoregressive parameters are obtained.

A hybrid dynamic forecast model for analyzing celebrity endorsement effects on consumer attitudes

This study investigates the time-varying effects of celebrity endorsements on consumer purchase attitudes toward promoted products using a novel dynamic hierarchical multi-attribute attitude forecast model. The induced direct and indirect effects via constructs of product attributes and net product value are then incorporated into the proposed conceptual model, which is formulated with a discrete-time nonlinear stochastic system. An empirical study of product categories of sport shoes and notebook computers demonstrates the feasibility of the proposed methodology. The analytical results demonstrate the capability of the proposed model to forecast consumer attitudes toward a promoted product and reveal the potential heterogeneity in patterns of attitude changes characterized by product attributes, price, and endorser performance as perceived by consumers. Furthermore, we infer that celebrity endorsement can significantly influence consumer purchase attitudes via both direct and indirect effects through product-attribute construct.

Optimal Filters with Multiple Packet Losses and its Application in Wireless Sensor Networks

This paper is concerned with the filtering problem for both discrete-time stochastic linear (DTSL) systems and discrete-time stochastic nonlinear (DTSN) systems. In DTSL systems, an linear optimal filter with multiple packet losses is designed based on the orthogonal principle analysis approach over unreliable wireless sensor networks (WSNs), and the experience result verifies feasibility and effectiveness of the proposed linear filter; in DTSN systems, an extended minimum variance filter with multiple packet losses is derived, and the filter is extended to the nonlinear case by the first order Taylor series approximation, which is successfully applied to unreliable WSNs. An application example is given and the corresponding simulation results show that, compared with extended Kalman filter (EKF), the proposed extended minimum variance filter is feasible and effective in WSNs.

Robust ℋ2 sliding mode control for non-linear discrete-time stochastic systems

This study is concerned with the sliding mode control (SMC) problem for uncertain non-linear discrete-time stochastic systems with H 2 performance constraints. The non-linearities considered are of both the matched and unmatched forms, which can encompass several classes of well-studied non-linearities. A new discrete-time switching function is proposed and then a sufficient condition is derived which ensures both the exponentially mean-square stability and the H 2 performance in the sliding surface. Such a condition is expressed as the feasibility of a set of linear matrix inequalities (LMIs) with an equality constraint. An improved discrete-time reaching condition is established to obtain the required SMC controller capable of driving the state trajectories onto the specified switching surface, along which then resulting in a non-increasing zigzag-type sliding motion. A simulation example is given to show the effectiveness of the proposed method.

Recursive estimation of discrete-time signals from nonlinear randomly delayed observations

In this paper, one-stage prediction, filtering, and fixed-point smoothing problems are addressed for nonlinear discrete-time stochastic systems with randomly delayed measurements perturbed by additive white noise. The observation delay is modelled by a sequence of independent Bernoulli random variables whose values–zero or one–indicate that the real observation arrives on time or it is delayed one sampling time and, hence, the available measurement to estimate the signal is not updated. Assuming that the state–space model generating the signal to be estimated is unknown and only the covariance functions of the processes involved in the observation equation are available, recursive estimation algorithms based on linear approximations of the real observations are proposed.

Derivative-free estimation methods: New results and performance analysis

The derivative-free nonlinear estimation methods exploiting the Stirling’s interpolation and the unscented transformation for discrete-time nonlinear stochastic systems are treated. The divided difference and unscented filters, smoothers, and predictors based on the methods are introduced in the unified framework. The new relations among the first order Stirling’s interpolation, the second order Stirling’s interpolation, and the unscented transformation are derived and their impact on the covariance matrices of the state estimates of the corresponding filters is analysed. The theoretical results are illustrated and used for the explanation of the unexpected behaviour of the sigma point Gaussian sum filters given as a mixture of the derivative-free filters.